Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Hector Laos

doi:10.1007/978-3-030-75914-8_5

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Hector Laos

Mechanical Systems Modeling

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

Original language	English
Title of host publication	Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
Editors	David S. Epp
Publisher	Springer
Pages	39-54
Number of pages	16
ISBN (Print)	9783030759131
DOIs	https://doi.org/10.1007/978-3-030-75914-8_5
State	Published - 2022
Event	39th IMAC, A Conference and Exposition on Structural Dynamics, 2021 - Virtual, Online Duration: Feb 8 2021 → Feb 11 2021

Publication series

Name	Conference Proceedings of the Society for Experimental Mechanics Series
ISSN (Print)	2191-5644
ISSN (Electronic)	2191-5652

Conference

Conference	39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
City	Virtual, Online
Period	02/8/21 → 02/11/21

Funding

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan).

Keywords

Equations of motion
Generalized forces
Nonlinear
Rigid body
Rotordynamics

Access to Document

10.1007/978-3-030-75914-8_5

Cite this

Laos, H. (2022). Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases. In D. S. Epp (Ed.), Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021 (pp. 39-54). (Conference Proceedings of the Society for Experimental Mechanics Series). Springer. https://doi.org/10.1007/978-3-030-75914-8_5

Laos, Hector. / Equations of Motion for the Vertical Rigid-Body Rotor : Linear and Nonlinear Cases. Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. editor / David S. Epp. Springer, 2022. pp. 39-54 (Conference Proceedings of the Society for Experimental Mechanics Series).

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title = "Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases",

abstract = "Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.",

keywords = "Equations of motion, Generalized forces, Nonlinear, Rigid body, Rotordynamics",

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Laos, H 2022, Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases. in DS Epp (ed.), Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. Conference Proceedings of the Society for Experimental Mechanics Series, Springer, pp. 39-54, 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021, Virtual, Online, 02/8/21. https://doi.org/10.1007/978-3-030-75914-8_5

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases. / Laos, Hector.
Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. ed. / David S. Epp. Springer, 2022. p. 39-54 (Conference Proceedings of the Society for Experimental Mechanics Series).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

AB - Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

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Laos H. Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases. In Epp DS, editor, Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. Springer. 2022. p. 39-54. (Conference Proceedings of the Society for Experimental Mechanics Series). doi: 10.1007/978-3-030-75914-8_5

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Abstract

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Keywords

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